Fractional diffusion models of option prices in markets with jumps

Álvaro Cartea; Diego del-Castillo-Negrete

doi:10.1016/j.physa.2006.08.071

Fractional diffusion models of option prices in markets with jumps

Álvaro Cartea, Diego del-Castillo-Negrete

Plasma Theory and Modeling

Research output: Contribution to journal › Article › peer-review

259 Scopus citations

Abstract

Most of the recent literature dealing with the modeling of financial assets assumes that the underlying dynamics of equity prices follow a jump process or a Lévy process. This is done to incorporate rare or extreme events not captured by Gaussian models. Of those financial models proposed, the most interesting include the CGMY, KoBoL and FMLS. All of these capture some of the most important characteristics of the dynamics of stock prices. In this article we show that for these particular Lévy processes, the prices of financial derivatives, such as European-style options, satisfy a fractional partial differential equation (FPDE). As an application, we use numerical techniques to price exotic options, in particular barrier options, by solving the corresponding FPDEs derived.

Original language	English
Pages (from-to)	749-763
Number of pages	15
Journal	Physica A: Statistical Mechanics and its Applications
Volume	374
Issue number	2
DOIs	https://doi.org/10.1016/j.physa.2006.08.071
State	Published - Feb 1 2007

Funding

The authors are grateful to Pablo Padilla for useful discussions on the applications of fractional calculus in finance. Cartea acknowledges financial support from the Nuffield Foundation NAL/00791/G.

Funders	Funder number
Nuffield Foundation NAL/00791/G

Keywords

Barrier options
CGMY
Double knock-out
Down-and-out
FMLS
Fractional calculus
Fractional-Black-Scholes
KoBoL
Lévy-stable processes
Riemann-Liouville fractional derivative
Up-and-out

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10.1016/j.physa.2006.08.071

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title = "Fractional diffusion models of option prices in markets with jumps",

abstract = "Most of the recent literature dealing with the modeling of financial assets assumes that the underlying dynamics of equity prices follow a jump process or a L{\'e}vy process. This is done to incorporate rare or extreme events not captured by Gaussian models. Of those financial models proposed, the most interesting include the CGMY, KoBoL and FMLS. All of these capture some of the most important characteristics of the dynamics of stock prices. In this article we show that for these particular L{\'e}vy processes, the prices of financial derivatives, such as European-style options, satisfy a fractional partial differential equation (FPDE). As an application, we use numerical techniques to price exotic options, in particular barrier options, by solving the corresponding FPDEs derived.",

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Fractional diffusion models of option prices in markets with jumps

Abstract

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Keywords

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